Connecting engagement and focus in pedagogic task design

Abstract

Teachers of mathematics face a particular tension, which the authors call the planning paradox. If teachers plan from objectives, the tasks they set are likely to be unrewarding for the pupils and mathematically impoverished. Planning from tasks may increase pupils' engagement but their activity is likely to be unfocused and learning difficult to assess. By seeking inspiration from research in the areas of curriculum design, the nature of authenticity in the classroom and the use of tools, and by looking retrospectively at the design of computer-based tasks that have underpinned their research for many years, the authors recognise a theme of purposeful activity, leading to a planned appreciation of utilities for certain mathematical concepts. The authors propose utility as a third dimension of understanding, which can be linked to purpose in the effective design of tasks. The article concludes with a set of heuristics to guide such planning.

Mathematical tasks and the student: navigating “tensions of intentions” between designers, teachers, and students

“…By taking into account students' voice and agency, researchers have offered task design constructs and principles that suggest ways in which task design and implementation may focus on students' task experiences (e.g., Ainley, Pratt, & Hansen, 2006;Bikner-Ahsbahs & Janßen, 2013;Clarke & Mesiti, 2013;Coles & Brown, 2013). Ainley, Pratt, and Hansen (2006) introduced the constructs of task purpose and utility, defining a task's purpose and utility from the perspective of the student rather than the task designer.…”

“…Ainley, Pratt, and Hansen (2006) introduced the constructs of task purpose and utility, defining a task's purpose and utility from the perspective of the student rather than the task designer. Clarke and Mesiti (2013) showed how teachers could enact tasks that build on students' real-time responses.…”

Supporting Mathematics Coaches’ Learning of Probability through Professional Development Tasks

“…This may include characteristics of tasks for learning (Hiebert et al, 1997;Stein, Smith, Henningsen, & Silver, 2000), the relationship between tasks and mathematical discourse (Boaler, 2008;Smith & Stein, 2011), and the role of task implementation and learning (Ainley, Pratt, & Hansen, 2006;Berry, Bol, & McKinney, 2009;Doerr, 2006;Henningsen & Stein, 1997). Van de Walle et al (2013) described features of problems for learning where learners must perceive such problems as "problematic" (p. 35).…”

Scaffolding norms of argumentation-based inquiry in a primary mathematics classroom

“…Research has pointed to problems with forms of inquiry, for example those which focus on activity but not depth of content (Artigue & Blomhøj, 2013) or on content without a sense of purpose and utility of that content (Ainley, Pratt & Hansen, 2006) or on open-ended inquiry which lacks support and guidance (Kirschner, Sweller & Clark, 2006). To address these problems, we have included the development of norms within lessons rich in purposeful and useful mathematical content and emphasised the teacher's role in scaffolding these norms.…”